Gap Sequence And Spectrum Of Cutting Sequence

Cutting sequence, as a kind of aperiodic sequences with minimal language complexity, have been studied for a long time in many aspects of mathematics, physics, computer science, etc. These sequences appear under many ditterent names, such as Sturmian sequences, rotation sequences, Christottel words, Beatty sequences, characteristic sequences, balanced sequences, and so forth.In this paper, we consider the properties of gap sequences of three typical cutting sequences. Let ω be a factor, then it occurs in the sequence infinitely many times. Letωpbe the p-th occurrence of ω and Gp(ω) be the gap between ωpand ωp+1. We define the kernel words and give the decomposition of each factor ω with respect to its kernel.They are the main tools to discuss the properties of gap sequences. Using the kernel and decomposition, we prove that:(1)Any factor ω of Fibonacci sequence has exactly two distinct gaps G1(ω) and G2(ω); the gap sequence {Gp(ω)} is the Fibonacci sequence over the alphabet {G1(ω), G2(ω)}.(2)Any factor ω of Tribonacci sequence has exactly three distinct gaps G1(ω), G2(ω) and G4(ω); the gap sequence {Gp(ω)} is the Tribonacci sequence over the alphabet {G1(ω), G2(ω), G4(ω)}.(3)Any factor ω of Fd,∞has exactly two distinct gaps G1(ω) and GB(ω); the gap sequence {Gp(ω)} is the sequence S over the alphabet {G1(ω), GB(ω)}, where B and S are only depending on the type of the kernel word of ω. The methods to discuss gap sequences of the three typical cutting sequences are ditterent.As an application, we introduce the spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors. Using the results above,we determine completely the spectrums.